[ENH] Port MSM (Move-Split-Merge) Centroids/Averages to Aeon

[Adityakushwaha2006]

Describe the feature or idea you want to propose

I am opening this issue to track a feature request suggested by @TonyBagnall on Discord.

He proposed porting the MSM Mean (barycenter) implementation into aeon to expand our clustering capabilities. Currently, we support the MSM metric, but without a centroid/mean function, it cannot be effectively used in centroid-based clustering algorithms like K-Means.

Describe your proposed solution

I plan to execute this port based on the reference implementation Tony provided: JanaHolznigenkemper/msm_mean.

My implementation plan is:

Adapt the Logic: Port the core barycenter algorithm to fit aeon's clustering.averaging module structure.

Optimize: Ensure the iterative update steps utilize numba to meet aeon's performance standards.

Validate: Add tests to ensure the calculated mean is correct and consistent with the original implementation.

Describe alternatives you've considered, if relevant

No response

Additional context

Happy to have others on board and collaborate on this feature additon : )
Please ping me in the comments if any contributor wants to pitch in.

[Adityakushwaha2006]

@aeon-actions-bot assign @Adityakushwaha2006

[satwiksps]

Thanks for creating an issue to formally track the feature @Adityakushwaha2006 . I have been working this on since last week. Feel free to suggest any improvements, if you have any once you are done with other works later this week.

[Adityakushwaha2006]

Hey @satwiksps , Just went through your implementation, it seems really well optimized.
This seems expansively complete, really commendable work

Ill unassign myself from this issue then
@aeon-actions-bot unassign @Adityakushwaha2006

Let me know if you need any help with anything else that might come up

[aeon-actions-bot]

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